Welcome Zoglchaim:
I think you need to take the fixture down and check the wiring to each bulb. One or more of them are wired in series rather than paralell. It can be remedied by making sure all black wires are together and all white wires are together. Make sure you run a bare copper ground wire too. Not using a ground wire could mess it up comming off the existing GFCI.
Let us know what you find.
Glenn

since you have added wall lighting fixture with on/off switch located directly on the fixture, there might be some wrong connection in wire between the connections of 6 bulbs that why when your switch on a light its dimming. Then while unscrewing a bulb rest others are lighting on. better check the connected wire between the bulbs?